Subgroups — C₂³×C₄×Q₈×C₃₇ #22683896

Order 9472.

12446 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₃₇	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₄×C₃₇	C₂	1	3	yes	maximal
4	C₂³×C₄²×C₃₇	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₃₇	C₄	1	32	yes	direct product factor
6	C₂³×Q₈×C₃₇	C₂²	1	28	yes	—
7	C₂⁴×C₄×C₃₇	C₂²	1	3	yes	—
8	C₂⁴×C₄×C₃₇	C₂²	1	3	yes	—
9	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₇	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊C₄×C₃₇	C₂²	1	84	yes	—
12	C₂²×C₄²×C₃₇	C₂²	1	84	yes	—
13	C₂⁵×C₃₇	C₂³	1	1	yes	characteristic
14	C₂²×Q₈×C₃₇	C₂×C₄	1	448	yes	direct product factor
15	C₂²×Q₈×C₃₇	C₂³	1	112	yes	—
16	C₂³×C₄×C₃₇	—	2	24	no	—
17	C₂³×C₄×C₃₇	C₂×C₄	1	48	yes	—
18	C₂³×C₄×C₃₇	Q₈	1	16	yes	direct product factor
19	C₂³×C₄×C₃₇	C₂³	1	42	yes	—
20	C₂³×C₄×C₃₇	C₂³	1	42	yes	—
21	C₂³×C₄×C₃₇	C₂³	1	14	yes	—
22	C₄×Q₈×C₃₇	C₂³	1	512	yes	direct product factor
23	C₂×C₄⋊C₄×C₃₇	C₂³	1	336	yes	—
24	C₂×C₄²×C₃₇	C₂³	1	336	yes	—
25	C₂⁴×C₃₇	C₂²×C₄	1	8	yes	—
26	C₂⁴×C₃₇	C₂×Q₈	1	8	yes	—
27	C₂⁴×C₃₇	#41	1	8	yes	—
28	C₂⁴×C₃₇	C₂⁴	1	7	yes	—
29	C₂×Q₈×C₃₇	C₂²×C₄	1	896	yes	direct product factor
30	C₂×Q₈×C₃₇	C₂⁴	1	64	yes	—
31	C₂²×C₄×C₃₇	—	2	168	no	—
32	C₂²×C₄×C₃₇	C₂²×C₄	1	336	yes	—
33	C₂²×C₄×C₃₇	C₂×Q₈	1	112	yes	direct product factor
34	C₂²×C₄×C₃₇	C₂⁴	1	84	yes	—
35	C₂²×C₄×C₃₇	C₂⁴	1	84	yes	—
36	C₂²×C₄×C₃₇	C₂⁴	1	28	yes	—
37	C₄⋊C₄×C₃₇	C₂⁴	1	192	yes	—
38	C₄²×C₃₇	C₂⁴	1	192	yes	—
39	C₂³×C₃₇	C₄×Q₈	1	64	yes	direct product factor
40	C₂³×C₃₇	C₂³×C₄	1	28	yes	—
41	C₂³×C₃₇	C₂²×Q₈	1	28	yes	—
42	C₂³×C₃₇	C₂×#41	1	28	yes	—
43	C₂³×C₃₇	C₂⁵	1	7	yes	—
44	Q₈×C₃₇	C₂³×C₄	1	256	yes	direct product factor
45	C₂×C₄×C₃₇	—	2	168	no	—
46	C₂×C₄×C₃₇	C₂³×C₄	1	336	yes	—
47	C₂×C₄×C₃₇	C₂²×Q₈	1	112	yes	direct product factor
48	C₂×C₄×C₃₇	C₂⁵	1	24	yes	—
49	C₂×C₄×C₃₇	C₂⁵	1	24	yes	—
50	C₂×C₄×C₃₇	C₂⁵	1	8	yes	—
51	C₂³×C₄×Q₈	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₃₇	C₂×C₄×Q₈	1	112	yes	direct product factor
53	C₂²×C₃₇	C₂⁴×C₄	1	14	yes	—
54	C₂²×C₃₇	C₂³×Q₈	1	14	yes	—
55	C₂²×C₃₇	C₂²×#41	1	14	yes	—
56	C₂²×C₃₇	C₂⁶	1	1	yes	characteristic
57	C₄×C₃₇	—	2	24	no	—
58	C₄×C₃₇	C₂⁴×C₄	1	48	yes	—
59	C₄×C₃₇	C₂³×Q₈	1	16	yes	direct product factor
60	C₂⁴×Q₈	C₂×C₃₇	1	1	yes	characteristic
61	C₂²×C₄×Q₈	C₂×C₃₇	1	56	yes	direct product factor
62	C₂³×C₄⋊C₄	C₂×C₃₇	1	3	yes	—
63	C₂³×C₄²	C₂×C₃₇	1	3	yes	—
64	C₂×C₃₇	C₂²×C₄×Q₈	1	28	yes	direct product factor
65	C₂×C₃₇	C₂⁵×C₄	1	1	yes	characteristic
66	C₂×C₃₇	C₂⁴×Q₈	1	1	yes	characteristic
67	C₂×C₃₇	C₂³×#41	1	1	yes	characteristic
68	C₂³×Q₈	C₄×C₃₇	1	32	yes	direct product factor
69	C₂³×Q₈	C₂²×C₃₇	1	28	yes	—
70	C₂⁴×C₄	C₂²×C₃₇	1	3	yes	—
71	C₂⁴×C₄	C₂²×C₃₇	1	3	yes	—
72	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
73	C₂×C₄×Q₈	C₂²×C₃₇	1	448	yes	direct product factor
74	C₂²×C₄⋊C₄	C₂²×C₃₇	1	84	yes	—
75	C₂²×C₄²	C₂²×C₃₇	1	84	yes	—
76	C₃₇	C₂³×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
77	C₂⁵	C₂³×C₃₇	1	1	yes	characteristic
78	C₂²×Q₈	C₂×C₄×C₃₇	1	448	yes	direct product factor
79	C₂²×Q₈	C₂³×C₃₇	1	112	yes	—
80	C₂³×C₄	—	2	24	no	—
81	C₂³×C₄	C₂×C₄×C₃₇	1	48	yes	—
82	C₂³×C₄	Q₈×C₃₇	1	16	yes	direct product factor
83	C₂³×C₄	C₂³×C₃₇	1	42	yes	—
84	C₂³×C₄	C₂³×C₃₇	1	42	yes	—
85	C₂³×C₄	C₂³×C₃₇	1	14	yes	—
86	C₄×Q₈	C₂³×C₃₇	1	512	yes	direct product factor
87	C₂×C₄⋊C₄	C₂³×C₃₇	1	336	yes	—
88	C₂×C₄²	C₂³×C₃₇	1	336	yes	—
89	C₂⁴	C₂²×C₄×C₃₇	1	8	yes	—
90	C₂⁴	C₂×Q₈×C₃₇	1	8	yes	—
91	C₂⁴	#41×C₃₇	1	8	yes	—
92	C₂⁴	C₂⁴×C₃₇	1	7	yes	—
93	C₂×Q₈	C₂²×C₄×C₃₇	1	896	yes	direct product factor
94	C₂×Q₈	C₂⁴×C₃₇	1	64	yes	—
95	C₂²×C₄	—	2	168	no	—
96	C₂²×C₄	C₂²×C₄×C₃₇	1	336	yes	—
97	C₂²×C₄	C₂×Q₈×C₃₇	1	112	yes	direct product factor
98	C₂²×C₄	C₂⁴×C₃₇	1	84	yes	—
99	C₂²×C₄	C₂⁴×C₃₇	1	84	yes	—
100	C₂²×C₄	C₂⁴×C₃₇	1	28	yes	—
101	C₄⋊C₄	C₂⁴×C₃₇	1	192	yes	—
102	C₄²	C₂⁴×C₃₇	1	192	yes	—
103	C₂³	C₄×Q₈×C₃₇	1	64	yes	direct product factor
104	C₂³	C₂³×C₄×C₃₇	1	28	yes	—
105	C₂³	C₂²×Q₈×C₃₇	1	28	yes	—
106	C₂³	C₂×#41×C₃₇	1	28	yes	—
107	C₂³	C₂⁵×C₃₇	1	7	yes	—
108	Q₈	C₂³×C₄×C₃₇	1	256	yes	direct product factor
109	C₂×C₄	—	2	168	no	—
110	C₂×C₄	C₂³×C₄×C₃₇	1	336	yes	—
111	C₂×C₄	C₂²×Q₈×C₃₇	1	112	yes	direct product factor
112	C₂×C₄	C₂⁵×C₃₇	1	24	yes	—
113	C₂×C₄	C₂⁵×C₃₇	1	24	yes	—
114	C₂×C₄	C₂⁵×C₃₇	1	8	yes	—
115	C₂²	C₂×C₄×Q₈×C₃₇	1	112	yes	direct product factor
116	C₂²	C₂⁴×C₄×C₃₇	1	14	yes	—
117	C₂²	C₂³×Q₈×C₃₇	1	14	yes	—
118	C₂²	C₂²×#41×C₃₇	1	14	yes	—
119	C₂²	C₂⁶×C₃₇	1	1	yes	characteristic
120	C₄	—	2	24	no	—
121	C₄	C₂⁴×C₄×C₃₇	1	48	yes	—
122	C₄	C₂³×Q₈×C₃₇	1	16	yes	direct product factor
123	C₂	C₂²×C₄×Q₈×C₃₇	1	28	yes	direct product factor
124	C₂	C₂⁵×C₄×C₃₇	1	1	yes	characteristic
125	C₂	C₂⁴×Q₈×C₃₇	1	1	yes	characteristic
126	C₂	C₂³×#41×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄×C₃₇	C₂²
commutator	C₂	C₂⁵×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁶×C₃₇
socle, prime radical	C₂⁵×C₃₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₃₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₃₇	C₂	direct (N × H)
5. C₂³×Q₈×C₃₇	C₄	direct (N × H)
10. C₂×C₄×Q₈×C₃₇	C₂²	direct (N × H)
14. C₂²×Q₈×C₃₇	C₂×C₄	direct (N × H)
18. C₂³×C₄×C₃₇	Q₈	direct (N × H)
22. C₄×Q₈×C₃₇	C₂³	direct (N × H)
29. C₂×Q₈×C₃₇	C₂²×C₄	direct (N × H)
33. C₂²×C₄×C₃₇	C₂×Q₈	direct (N × H)
39. C₂³×C₃₇	C₄×Q₈	direct (N × H)
44. Q₈×C₃₇	C₂³×C₄	direct (N × H)
47. C₂×C₄×C₃₇	C₂²×Q₈	direct (N × H)
51. C₂³×C₄×Q₈	C₃₇	direct (N × H)
52. C₂²×C₃₇	C₂×C₄×Q₈	direct (N × H)
59. C₄×C₃₇	C₂³×Q₈	direct (N × H)
61. C₂²×C₄×Q₈	C₂×C₃₇	direct (N × H)
64. C₂×C₃₇	C₂²×C₄×Q₈	direct (N × H)
68. C₂³×Q₈	C₄×C₃₇	direct (N × H)
73. C₂×C₄×Q₈	C₂²×C₃₇	direct (N × H)
76. C₃₇	C₂³×C₄×Q₈	direct (N × H)
78. C₂²×Q₈	C₂×C₄×C₃₇	direct (N × H)
82. C₂³×C₄	Q₈×C₃₇	direct (N × H)
86. C₄×Q₈	C₂³×C₃₇	direct (N × H)
93. C₂×Q₈	C₂²×C₄×C₃₇	direct (N × H)
97. C₂²×C₄	C₂×Q₈×C₃₇	direct (N × H)
103. C₂³	C₄×Q₈×C₃₇	direct (N × H)
108. Q₈	C₂³×C₄×C₃₇	direct (N × H)
111. C₂×C₄	C₂²×Q₈×C₃₇	direct (N × H)
115. C₂²	C₂×C₄×Q₈×C₃₇	direct (N × H)
122. C₄	C₂³×Q₈×C₃₇	direct (N × H)
123. C₂	C₂²×C₄×Q₈×C₃₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×Q₈×C₃₇	2
C₂²×C₄×Q₈×C₃₇	2
C₂³×C₄⋊C₄×C₃₇	2
C₂³×C₄²×C₃₇	2
C₂³×C₄×Q₈	37

Subgroups — C23×C4×Q8×C37 #22683896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×Q₈×C₃₇ #22683896